In this paper, we deal with the spacelike linear Weingarten submanifolds with parallel normalized mean curvature vector in an $$(n+p)$$ -dimensional semi-Riemannian space form $$N^{n+p}_{q}(c)$$ of constant sectional curvature c with index q, where $$1\le q\le p$$ . In this setting, we obtain an important inequality and apply some appropriated generalized maximum principles to a suitable Cheng–Yau-modified operator to obtain some characterizations of the linear Weingarten submanifolds.